Can renewables meet our growing demands? We think so - take a look at our technical report
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Can renewables meet our growing demands? We think so - take a look at our technical report Can renewables meet our growing demands? We think so - take a look at our technical report Document Transcript

  • Integrated Renewable Energy Systems: Initial Technical Report. copyright ⃝2010 Noel McWilliam, Mitravitae. c June 2011Disclaimer ond year gives rise to a further 25.9% increase in wa- ter production, a 5.3% increase in grain yield andThe information provided herein is intended for in- compost production and a 13% increase in electricityformational purposes only, Mitravitae will not be li- production. Year three corresponds to near equilib-able for any errors, omissions, or delays in this in- rium conditions with biogas production permittingformation or any losses, injuries, or damages arising nominal overnight operation of the plant throughoutfrom its display or use. the year. Secondary benefits may include:Abstract • Carbon abatement — bio-digestion of crop residues releases less CO2 into the atmosphereIn this paper we consider a novel integrated sys- than burning. Electricity production using solartem for the combined purpose of energy generation, and biofuel replaces fossil-fuel systems.salt-water desalination and biofuel production. Ourconfiguration combines complementary sub-systemswhereby outputs from one system are used as in- • Ground water improvements — usingputs to another creating a positive feedback mech- sea/brackish water as a secondary sourceanism which significantly increases system efficiency for irrigation and drinking water may helpand output. This mechanism is not present when the alleviate pressure and perhaps even rechargecomponents are considered separately. dwindling fresh water reserves in arid and We evidence our claim through analysis of a semi-arid areas.demonstration configuration that combines math-ematical models and data for Concentrating So- • Land regeneration — composting bio-digestionlar Thermal (CST), Unfired Boiler Steam Genera- by-products and irrigation using water havingtion, Multi-Stage Flash (MSF), wheat-crop produc- low salt content increases humic and nutrienttion and bio-digestion which together comprise an ex- content and reduces soil salinity.ample integrated energy system. Industry standardmodels are used and adapted where necessary — full • Matching supply with demand — the best loca-details are given and all data are fully referenced. tions for supply of solar energy also tend to be We consider system performance for a site near hot and arid having low population densities andAhmedabad, Gujarat, India. In the first year alone low demand. By producing both food and wateruse of desalted water for irrigation results in a 40% the system provides fundamental essentials forincrease in crop yield and compost production over economic regeneration and thereby energy de-baseline levels. The feedback mechanism in the sec- mand in such areas. 1
  • Introduction omission since as we demonstrate herein there are features unique to our set-up that do not exist whenThe transient nature of energy generation from re- considered in isolation:newable sources is problematic in that supply is mostoften asynchronous to demand. This is particularly • closed autonomous production — outputs of oneacute in power production since it is not generally sub-system are inputs to another, no extraneousfeasible to store electrical power, and so highly vari- fossil energy input required,able demand must be met with near instantaneouschanges in generation. Unlike fossil-fuel (typically • positive feedback — by completing a closed-loopgas) technologies that can be brought on-line rapidly, energy circuit we markedly increase overall sys-nuclear and renewable production is relatively steady tem efficiency thereby increasing output,and cannot therefore complete the market in peak • natural hedge against insolence variation — lowtimes unless capacity over-supplies at other times. insolence levels negatively correlate with precip-Back-up energy sources are usually installed in solar itation. Otherwise reduced solar production andsystems and are often fossil-powered or make use of desalination rates are supported by increasedcostly molten nitrate salt technologies to store energy crop yields and hence biofuel availability result-for overnight use. ing from increased precipitation levels. The con- Geographic variation of natural resources also lim- verse also holds.its the potential of renewables to compete with fos-sils: solar energy on its own is a vast resource with In this paper we estimate the performance of our sys-ground-level global irradiance levels totalling about tem for an example configuration using existing tech-170 peta (1015 ) watts which is enough to meet the nologies. We demonstrate the existence of a positiveworld’s electricity needs tens of thousands of times feedback mechanism and quantify the benefits con-over. As one might expect, equatorial regions with ferred. In doing so we highlight many other advan-dry climates provide the best locations having mini- tages to our an integrated systems approach. In par-mum attenuation and little or no scattering of light ticular, by simultaneously addressing food and waterfrom atmospheric moisture. scarcity issues in semi-arid and arid regions we help to Unfortunately these areas also tend to be desert, remove limiting factors to energy demand that mightarid or semi-arid with low population densities and exist in areas where solar energy supply is greatest.low demand. Hence supply is high where demand is In the next section we give a definition of the com-low. The absence of capital-intensive extensive trans- position of our integrated system and highlight somemission infrastructures linking inhospitable regions to of the properties and advantages conferred. The re-areas of high demand is a significant barrier to real- mainder of the paper quantifies system outputs for anising the potential of solar. Although there is some example configuration. Wherever possible we makediscussion for large-scale systems in the Sahara and use of published models for each of the system com-Thar deserts, it seems that such costs will otherwise ponents. In the interests of clarity and transparencyexclude smaller scale installations in the best loca- we give full mathematical details whenever adapta-tions. tions or analysis not appearing in the literature are In this study we consider a novel system compris- three established renewable technologies that aretypically implemented singularly. While there are ex-amples of two-component prototype systems such as System Concept & Featuressolar co-generation, and hybrid solar power and bio-fuel systems, to the author’s knowledge there are no The system comprises three components — a solarexamples prototyped or in the literature that combine thermal power plant, a desalination plant and a bio-three systems. In our opinion this is a fundamental fuel generation and combustion unit. 2
  • In this report we concern ourselves with the pos- Extraneous Irrigation Source &sible use of desalinated water for irrigation of crops Groundwater Rechargeto the ultimate end of producing biogas from diges-tion of crop residue for use as fuel in a back-up boiler In this report we outline an example in which the so-supporting a solar thermal plant. There are many ad- lar thermal plant is supported by irrigation allowingvantages that arise by combining these three systems it to operate day and night throughout the year. Thisand we outline a few of these in the following: is achieved assuming only 60% of irrigation comes from rain/ground-water sources; thus almost twice as much water is input to the soil than is taken fromWater & Biomass as Energy Storage the ground, potentially recharging this dwindling re- source.Irrigation water and the biomass supported by it actas energy storage where we transfer solar irradianceto water, to biomass, to biofuel and back to heat Atmospheric Pollution, Carbon Abate-and electricity when required. We do not use costly ment & Land Regenerationenergy storage mechanisms often employed in the Power from solar energy has clear benefits to the envi-overnight running of a plant, nor is there a reliance ronment and manifests itself here on several levels: iton fossil-fuel back-up systems or the infrastructure is common practice in India to burn crop residue, re-required to supply it. leasing large amounts of CO2 into the atmosphere. In contrast biogas combustion is the only source of CO2 in our configuration, with the majority of carbon con-Positive Feedback & Increased Effi- tained within the fermentation ‘by-product’ — whichciency makes quality compost. Composing increases soil nu- trient levels and humic content thereby sequesteringBy using outputs of the system as system inputs we carbon and improving the soil’s resistance to water-create a closed-loop energy circuit, setting-up a pos- logging and drought. An abundant availability of nu-itive feedback mechanism in which plant outputs are trient rich compost and irrigation water having lowincreased year-on-year. This is possible since each contaminant levels actively regenerates impoverishedunit output water used for irrigation increases biogas soils in semi-arid regions and could bring low-gradesupport the plant receives, which in turn increases disused land into production.the amount of water available for irrigation in the Land regeneration, groundwater recharge, food,following season. One might naturally suspect in- drinking water and power production — these bene-efficiencies in conversion and storage may result in fits are all procured using renewable sources, withoutrapidly diminishing returns at each iteration. While recourse to extraneous (fossil-fuel or otherwise) in-this is ultimately true, the use of water in crop pro- put and the associated supply and price risk therein.duction increases yield (by reducing water-stress and The efficiencies achieved are a consequence of positiveimproving plant photosynthetic efficiencies) thereby feedback mechanisms between the components, eachmaking available an additional source of solar energy component itself being an established technology into the solar thermal plant.1 In such instances energy its own right therefore posing limited technologicalinflux rates may actually increase at each subsequent and operational risks.iteration so that in some operational region the pos-itive feedback mechanism may re-enforce itself. 1 Efficiency may be also improved if initial support levels An Example Configurationare insufficient to ensure operation during daylight hours, sothat each increment in support increases the amount of energy We consider as an illustrative example a system com-that may be collected. prising a heliostat array and solar tower collection 3
  • system, an unfired boiler, a secondary gas boiler, a The pressure in each chamber successively reduces sosteam turbine and a multi-stage flash (MSF) desali- as to ensure the temperature of the heated brine en-nation plant which also serves as a condensing unit tering the chamber is above its boiling point, therebyfor the operation of the turbine. causing the brine to ‘flash’ (vapourise). As the brine In this set-up the flux of incoming solar energy is flashes it loses energy, necessitating a pressure-dropreflected by a set of flat-plate mirrors a. onto a re- in the next stage for the flashing process to continue.ceiver drum b. situated at the top of a tower c. around The steam produced is condensed on a tube bundlewhich the mirrors are distributed. The drum contains situated at the top of each chamber through whichwithin it a heat transfer fluid (HTF) which is heated cold brine flows, thus pre-heating it so that energyby the light incident upon the drum. This fluid is used to vapourise the brine is reclaimed. Reclaimingpumped to a heat exchanger in the unfired boiler d.. the latent heat of the flashing brine leads to muchA secondary feed-flow (FF) fluid that supplies the reduced energy requirements, and higher productionturbine is passed through the boiler and heat is trans- rates than simply boiling the brine.ferred from the HTF to the FF. If sufficient energy In the following subsections we detail the mathe-is available the FF is vapourised and the resulting matical models for each of the components.steam is used to power the turbine e.. Unlike fossil fuel power plants, the primary energy Solar Irradiance Modelsource for solar varies diurnally and seasonally whichtranslates to a variable steam FF rate to the turbine. Irradiance (G), measured in W/m2 , is the rate atAs is common in energy plant design we include a which radiant energy is incident upon a surface ofsecondary gas boiler which in this analysis is used unit area. In order to obtain a good picture of sys-to support the FF from the unfired boiler to ensure tem performance throughout the year one must usethe FF input to the turbine is above some minimum. detailed site-specific irradiance information. Unfor-We use equilibrium performance statistics for turbine tunately such data are not readily available at gran-output as quoted in manufacturer’s specifications. ular time-scales and so it is often necessary to in- The splitter f. ensures that the correct flow from terpolate sparse data by modelling the physical andthe condenser g. feeds the unfired boiler with the re- climatic factors that affect it.maining sent to the secondary boiler h.. The output In this study we use annual direct beam dual-flows from the unfired and fired boiler and combined tracking insolence values at a site in Ahmadabad,in the mixer i. which feeds the turbine. The output Gujarat (latitude 23.07, longitude 72.63 and altitudeflow from the turbine is typically wet steam at a re- 55m) as given in Table 1 of Purohit (2010). Theduced temperature and pressure. Before it can be re- annual insolence value being 1816 kW h/m2 /yr orheated by the boilers it must first be condensed and equivalently Ia = 1816 × 1000 × 602 J/m2 /yr.recirculated by a pump. The condenser here forms To generate irradiance levels for a given time of daypart of the MSF desalination plant so that it serves and year we must scale the annual insolence to taketo condense and cool the FF as well as provide a into account atmospheric attenuation and climaticheat source for desalination which would otherwise seasonality. We do not factor variations due to theunused. It is the author’s belief that the applica- Earth’s eccentricity, wavelength dependent attenua-tion of co-generation for the purpose of desalination tion, refraction or angular dispersion of the incidentprovides a good fit as regions most suitable for so- beam (although we do later account for dispersion oflar thermal power generation also tend to be most in reflected beams).need of fresh water. The energy obtained from the heat of vapourisa- Daily and Seasonal Atmospheric Attenuationtion and any additional cooling of the FF is trans-fered via a heat exchanger to the MSF input brine As light enters the atmosphere its intensity is dimin-flow which is then passed to the flashing chambers j.. ished due to scattering and absorbtion. The greater 4
  • the air mass though which it passes the higher thedegree of attenuation. More precisely we define theextinction coefficient Ce to be the proportion of solarbeam intensity incident at ground level where Ce = e−mand m is the air mass fraction — the ratio of the massof atmosphere through which beam radiation passesto the mass it would pass if the sun were at zenith(i.e. directly overhead). This fraction principally varies according to the po-sition of the sun in the sky (which itself will dependon the time of day, the declination of the Earth’saxis and site latitude) and elevation. We make use ofthe empirical relationship for m given by Kasten andYoung (1989) in Duffie and Beckman: Figure 1: Atmospheric Attenuation Values e−0.0001184h m= cos(θz ) + 0.5057(96.080 − θz )−1.634 As a first approximation we make use of monthlywhere h is the site elevation and θz is the zenith angle solar insolation data taken from [28] quoted for agiven as similar site in Ahmedabad (longitude 72.56, latitude 23.03). These values embed within them the seasonal θz = cos(ϕ) cos(δ) cos(ω) + sin(ϕ) sin(δ), (1) variation of Ce at a course scale. Not wishing to double-count this effect we must impose the seasonalwith ϕ the site latitide, the declination variation as implied by the monthly data. To achieve ( ) this we first interpolate the data using a cubic-spline 284 + n δ = 23.45 sin 360 to obtain daily insolence values. We then normalise 365 the values so that they integrate to 1 — these rep-being the angular position of the sun at solar noon resent the proportion of the annual insolence falling(when the sun is on the local meridian) with respect within any given day. Each set of Ce values for anyto the equator, n ∈ [1, 365] being the day of year; and given day are then scaled so as to sum to the dailyω the angular displacement of the sun east or west of seasonal values. In this way the integral of the scaledthe local meridian due to the rotation of the Earth factors over some time interval on any given day rep-(approximately 15◦ per hour). resents the proportion of the annual total insolence Figure 1 shows the effect of this variation at our falling within that time. The integral over the en-site. tire year is therefore 1 and so to obtain the requisite dual-tracking irradiance levels we need only multiply by Ia .Environmental Seasonality Figure 2 (bottom) shows the cubic-spline fit to theThe extinction coefficient is a model based on the monthly data, which clearly depicts the affect of theEarth’s orientation to the sun and physical properties monsoon on the insolence levels. At first glance theof the atmosphere. It does not account for enhanced peak irradiance levels in Figure 2 (top) appear muchattenuation due to site-specific environmental factors sharper than in the monthly data. By constructionsuch as atmospheric particulate matter, humidity and the irradiance levels for each day integrate to the levelcloud cover which typically vary on a seasonal basis. implied from the seasonal data. However the distri- 5
  • plant will be limited by ground-level irradiance. Proximately, additional constraints hold since the flat mirrors must be angled to reflect the light to the re- ceiver thereby reducing the effective area of the helio- stat array. This is commonly referred to as the cosine effect, which varies according to the relative position of the heliostat to the receiver and the position of the sun. Heliostats may also block the path of reflected light from other heliostats or reduce the light inci- dent to others by shading thereby further reducing the amount of available energy to the receiver. Im- perfections in the mirror surface will cause the re- flected light to disperse even when the incident light 1.3 has little or no angular dispersion. Finally the mir- 1.2 rors and glass will also absorb some of the incident energy changing the spectral characteristics of the ra- 1.1 diation. weighting factor 1 In the following subsections we consider each of 0.9 these effects. 0.8 0.7 Cosine Effect 0.6 To capture the cosine effect in three dimensional 0.5 0 90 180 270 365 space we must first model the position of the sun day of year as a function of the time of day and year. Equation 1 gives the azimuth angle, from which we may deter-Figure 2: Top: Estimated Irradiance (G). Bottom: Nor- mine the angular elevation. We also need to the solarmalised Cubic-Spline Fit to Monthly Insolence Values azimuth γs — the angular displacement from south of the projection of beam radiation on the horizontal plane:bution of this energy during the course of the day is ( )a function of the Earth’s declination, so that for a γs = sign(ω) cos−1 cos(θz ) sin(ϕ) − sin(δ)fixed daily insolence shorter days give rise to sharper sin(θz ) cos(ϕ)irradiance peaks. Hence the affect of the physicalattenuation model still impacts both the daily and from Duffie and Beckman, where sign(·) ∈ {−1, 1},seasonal variation of irradiance. and |·| is the absolute value of the operand. Thus the pair (θz , γs ) completely define the direc- tion of the solar beam radiation. To compute theHeliostat Array And Tower effective area of a single heliostat we determine theThe central receiver design consists of a set of flat- normal subtended to its surface required to reflectplate mirrors (heliostats) that track the sun’s position the incident beam to the centre of the receiver drumthroughout the day and reflect the incident light onto xR ∈ R3 : Let xHi ∈ R3 denote the position of centrea receiving drum positioned at the top of a tower. of the ith heliostat. We direct the beam reflected at Ultimately the amount of energy available to the the mid-point of the heliostat to the mid-point of the 6
  • drum so that the reflected beam direction vector is xR − xHi vHi = . ∥ xR − xHi ∥ 120The incident beam at xHi is given by the zenith 100and azimuth angles, whose representation in euclid-ian space we denote ns . The normal to the heliostat 80 zenithplane nHi is simply the vector that bisects the angle 60subtended between ns and vHi that is 40 100 ns + v H i 20 nHi = . 0 2 0 (+)N−S(−) −100 −50 0 50 100 150 −100 200 250To find the position of the vertices we assume the he- (−)W−E(+)liostat pivots about the mid-point whereby only theplane azimuth γHi and zenith θHi may be modified. Figure 3: Illustration of a Heliostat Field layout with 36Hence we do not permit rotations about nHi . Let bhi elements. Yellow lines depict beam direction of incidentand bvi respectively denote horizontal and vertical light. Blue the reflected beam and green the heliostat nor-basis vectors for the heliostat plane, the local origin mals. The red asterisk shows the position of the receiverbeing the mid-point. Their values are thus mid-point xR .   cos(γHi ) wHi   bhi = 0 2 The solution is therefore − sin(γHi ) [ ] −λand = M −1 xH j xS i  √1  cos(θHi ) sin(γHi ) hHi  2 where the inverse of the matrix M = [nS bs1 bs2 ] ∈ bvi = sin(γHi )  (2) 2 1 R3×3 exists by construction. √ cos(θH ) cos(γH ) 2 i i Repeating for all four vertices yields an the image of the heliostat as viewed by the sun whose area canwhere wHi and hHi are the width and height of the be determined by application of generic formulae forheliostat mirror. One may then easily determine the polyhedra. This is the effective area of the heliostat.positions xH j ; j = 1, ..., 4 of the four vertices as linear icombinations of the above basis vectors. Figure 3 shows the heliostat normals and incident Once we have the vertex positions we project them and reflected beams in a field of 36 heliostats ar-onto a ‘solar plane’ Ps — the plane perpendicular to ranged in a grid centred at the tower. Figure 5 topthe incident beam direction ns : we note that we may and bottom illustrate the magnitude of the cosinearbitrarily choose the basis bs1 and bs2 vectors for the effect during the course of a day (operating hourssolar plane so long as they are mutually perpendicular 6:00-18:00) for a west and north-facing heliostat re-to each other and ns . To project the j th vertex onto spectively. As the sun rises in the east the tower andthe plane we solve for λ ∈ R such that xH j + λnS ∈ sun are on opposite sides of the west-facing heliostat iPs . In matrix terms there must exist some λ and with the angle subtended between them being greaterxs ∈ R2 such that than 90◦ . In the period after the solar noon the an- gle is always acute giving rise to the observed skew. xH j + λnS = [bs1 bs2 ]xs . Also noteworthy is the shift in the peak values which i 7
  • 1 Shading and Blocking 10m 0.9 100m Thus far we have derived expressions for the basis 0.8 vectors and vertex positions of each heliostat. To 0.7 determine the effect of shading and blocking from a cosine factor 0.6 given heliostat we need only project the image of its 0.5 vertices onto the plane of another and calculate the 0.4 area of overlap of the two. The proportion of the area unshaded/unblocked is then used to scale the 0.3 reflected irradiance values for each heliostat. 0.2 In the case of shading all vertices are projected in 0.1 the direction of the solar beam ns onto the plane 0 0 6 12 18 24 of the shaded heliostat. Hence for the j th vertex of hour the ith heliostat we must find a λ such that xH j + i 1 λnS ∈ PHk — the plane of the k th heliostat. All 0.9 points on this plane are linear combinations of the 0.8 basis vectors bhk and bvk given in 2 which reference 0.7 xHk as their origin. The solution, as with the solar plane projections is thus cosine factor 0.6 [ ] 0.5 −λ 0.4 xi j = M −1 (xH j − xHk ) i H k 0.3 0.2 where now M = [nS bhk bvk ] with xi j ∈ R2 the H k 0.1 image of the j th vertex on the plane PHk . Note that 0 0 6 12 18 24 shadow is only cast in the region for which λ > 0. hour For blocking of the ith heliostat we must project its vertices onto the PHk and calculate the area of over-Figure 4: Cosine Effect: Effective Proportion of Area for lap. This is in contrast to shading where the overlapa West-Facing (top) and North-Facing (bottom) Heliostat corresponds to the shaded region of the k th heliostat.with Respect to Time of Day and Distance from Tower. In addition the projection is in the direction of the reflected beam vHi so that M = [vHi bhk bvk ]. While it is a relatively simple matter to determine the intersection of the image of the ith heliostat and the face of the k th , one must also account for shading and blocking that may occur from other heliostats. Overall the reflected image from the shaded/blockedoccur at the point where the sun moves as close as heliostat will be the complement of the union of allit can to being ‘behind’ the tower — after the solar such projections onto its surface. It is possible thennoon for a west-facing heliostat and before noon for that the reflection may compose an arbitrary numbereast-facing. Both effects become less pronounced as of disjoint polyhedra. Calculating these fragments isthe distance between the heliostat and tower is re- too computationally intensive for our purpose. Ourduced. In contrast the north-facing heliostat, which approach here is to calculate the overall shading andlies on the local meridian relative to the tower, has a blocking as a cumulative product of the proportionmore or less symmetrical view of the sun and neither of unshaded areas calculated for each projection. Ineffect is prominent here. this way we need only calculate the intersection of the 8
  • image and the face of the heliostat without reference and so clearlyto projections from other heliostats.2   −vHi (3) 1  . bRi h =√ 0Spillover and Dispersion vHi (1)2 + vHi (3)2 vHi (1)Spillover occurs when the reflected image from a he- Now that we have the receiver-plane bases we canliostat does not fall completely on the drum’s surface. project the vertices of the ith heliostat in the directionThis may be a significant loss if the heliostat is situ- of the reflected beam vHi . Again solutions are of theated at a distance from the receiver in which case an- formgular dispersion will increase the size of the reflected [ ] −λimage. Also if the heliostat is very close to the base = M −1 (xH j − xR ) xi j R iof the tower the angle subtended to the drum by thereflected beam may be large thereby ‘stretching’ the where M = [vHi bRi bR ] with xi j the projected image h vi Rimage. of the j th vertex on the receiver-plane. In this study we assume a cylindrical drum so that Thus far we have not accounted for dispersion. Infrom the view-point of each heliostat the receiver ap- reality the reflected beam will have an intensity dis-pears approximately as a flat plane tangential to the tribution associated with it due to scattering by im-drum surface in the horizonal x1 x3 -plane. As before perfections in the mirror and glass surface, refraction,our approach is to project an image of the heliostat absorbtion and emission and the angular dispersiononto this receiver-plane and determine the intersec- of the incident light. Bennett (2008) [19] models dis-tion. The overspill area is the proportion of the he- persion as a Gaussian distribution.liostat image falling outside of the receiver-plane. Since we are not per se interested in explicitly mod- For simplicity we will assume the plane origin is elling the intensity distribution on the drum (see thelocated at the centre of the drum xR . Before we can following section on the overall collection rate model)apply the projection we need expressions for the basis we adopt an approach in which we simply scale thevectors of the receiver-plane: we first note that the projected image: let L denote the length of one ofvertical basis bR is simply vi the sides of the projected image (before scaling). We   apply to the image the scaling factor 0 √ bR =  1  . vi L + 2dHi tan( α )2 F = 0 L where dHi = xHi − xR is the distance between theWe also note that since the reflected beam is directed receiver-plane and heliostat mid-points and α is thetoward the centre of the drum, the direction of the dispersion angle. Here we take α = 3 × 10−3 radiansnormal nR tangential to the plane is opposite to vHi i as suggested in Gstoehl [4].in the x1 x3 -plane — so that the plane ‘faces’ the he-liostat. Finally since the receiver-plane is vertical (onaccount of the drum being a cylinder) the normal has Heliostat Collection Rateno vertical component. Hence Together with the solar irradiance model, the he-   liostat losses derived herein determine the collection −vHi (1) ˙ 1   rate QH of solar energy received by the drum: nR = √ i 0 vHi (1)2 + vHi (3)2 −vHi (3) ˙ QH = ηf ield AH G 2 In effect we assume independence of the shaded regions. where G is the irradiance, AH the total perpendicularOne could readily create upper and lower bounds for the ef- area of the heliostat array andfects of shading/blocking by respectively assuming additivityor subadditivity of the shaded proportions. ηf ield = ηref lectance ηcos ηshading ηblocking ηspillover 9
  • Gee (1984) gives a simple formulation as 0.9 10m 0.8 20m Qc = FR QH − FR UL Ac (Tci − Ta ). ˙ ˙ 100m 0.7 Here the near linear dependencies (operating and en- 0.6 vironmental temperatures, drum area) are explicitly overspill 0.5 represented with the non-linear characteristics (wind 0.4 speed etc) lumped together as part of the overall 0.3 heat-loss coefficient UL . As a first approximation we assume a constant UL = 6.83 W/m2 /◦ C taken from 0.2 Table 1 in McMordie (1984). This value is for a so- 0.1 lar thermal tower with sodium and potassium nitrate 0 salt HFT operating on a similar temperature range 0 6 12 hour 18 24 Tco = 566◦ C, Tci = 288◦ C as will be considered here.Figure 5: Spillover vs Distance: Effective Proportion of Unfired BoilerArea for a West-Facing Heliostat with Respect to Time ofDay and Distance from Tower. Tower Height 61m. Drum In this configuration we make use of an unfired boiler5m. as a means to generate steam for use in the turbine from the heat collected in the receiver system. Here the HTF is pumped through a heat exchanger in thewith ηref lectance , ηcos , ηshading , ηblocking , ηspillover ∈ boiler into which feedwater enters and steam leaves.[0, 1] the scaling factors averaged over all ar- Usually the feedwater is injected into the boiler whereray elements respectively corresponding to the re- it mixes with water already there — thus the shellflectance efficiency, cosine effect, shading, blocking side is assumed to be uniformly at the steam sat-and spillover. uration temperature Tsat . Once the water reaches In this study we take the heliostat dimensions and saturation it is vapourised. The vapour then enters areflectance values given for the Psi 120 series in [16] superheater where the steam is further heated to thewhere ηref lectance = 0.93 with heliostat dimensions target turbine steam temperature Ts .12 × 10m. We consider an array of 750 elements ar- The thermal energy required to vapourise and su-ranged in a grid centered at the base of the tower. perheat the FF is supplied by the HTF output byA square (concentric to the array) exclusion zone of the solar collector and entering the heat exchanger40m was created so as to reduce overspill effects. at a temperature Tco . As it passes through it cools to the temperature Tci which is fed back as input to the collector. Because the boiler has a finite heatOverall Collection Rate Model exchange area, and the HTF a finite mass-flow rate ˙The rate of energy collection Qc is typically consid- ˙ Mc the temperature Tci exceeds the steam saturationered as the difference between the flux of incoming en- temperature. ˙ergy QH from the heliostat array and the losses due to The rate of energy transfer from the boiler to theconvection, conduction and radiation from the drum. FF is typically expressed as a function of the log-The amount of energy lost to the environment will be mean temperature differencea function of the operating temperature of the drum,drum characteristics (area, conductivity, emissivity Tc − Tci ∆Tm = oTc −Tsfor example), environmental conditions (ambient air ln Tco −Ts itemperature, wind speed, sky temperature) and op-erational parameters such as the HTF temperature. the area of the heat exchanger Ab and its heat trans- 10
  • fer coefficient Ub : Here we choose Ts to be as close as possible to a fixed target value TsT as determined from the turbine Ub Ab ∆Tm . specifications. At all times we ensure Ts ∈ [Ts , Ts ] lb ub remains within operational bounds. Originally weSince the energy absorbed by the FF is extracted had also fixed Fs and solved for Mc accordingly, how-from the HTF we have the energy balance equation ever this either led to unrealistically large HTF flow Mc cp (Tco − Tci ) = Ub Ab ∆Tm (3) rates at low irradiance levels or a markedly truncated operating time if an upper bound on Mc was imposed.where cp is the heat capacity of the HTF. If only Thus we consider Mc and TM SF as our control vari-some of the collected energy can be transferred to the ables with Fs our variable output.FF then the HTF temperature will increase, thereby Our strategy is to operate the unfired boiler withincreasing the heat-loss to the atmosphere. Hence in variable Fs and TM SF so as to maximise the timeequilibrium we have for which the heliostat array is operational. In this instance the MSF provides a regulatory function on Qc = Mc cp (Tco − Tci ) = Ub Ab ∆Tm . ˙ (4) Fs in its capacity to reduce TM SF (by altering the brine flow rate and temperature to the condenser)In our model we allow for the FF to enter the boiler which allows for a higher heat transfer rate to theat a temperature lower than the saturation temper- feed-flow thereby allowing the plant to remain withinature. The energy transferred from the HTF to the operational limits for longer periods.FF is thus The secondary gas fired boiler provides overnight Fs [∆Hsat + ∆Hv + ∆Hs ] = Mc cp (Tco − Tci ), (5) support to the plant at a specified overnight flow rate Fsupp so that the turbine is continuously operational.where ∆Hsat = Sw (Tsat − TM SF ) is the enthalpy To summarise our operational scheme:required to heat water to saturation temperature,∆Hv = (1 − ξ)hf g the enthalpy of vapourisation for • Fix Tci . Assume Ta fixed.the dry portion of the front-feed steam with ξ ∈ [0, 1] • If QH < FR UL Ac (Tci −Ta ) heat loss rate greater ˙the wet fraction of the turbine front-feed steam, and than flux — heliostat not operational. Set Fs =∆Hs = (ξSw + (1 − ξ)Sv )(Ts − Tsat ) the enthalpy 0, FM = Fsupp .of superheat for the superheated steam. Here Fs(kg/s) is the mass-flow rate of the FF to the unfired • HTF insulated: start-up temperature Tci .boiler, hf g the latent heat of vapourisation, Sw andSv the FF specific heat capacities of water and water • Fix turbine steam temperature Ts .vapour, and TM SF the temperature of the FF exitingthe multi-stage flash condenser. • Solve for Mc ∈ (Mc , Mc ] and TM SF ∈ lb ub These equations define the relationships between [TM SF , TM SF ] with Tco ∈ [Tco , Tco ] and Fs > 0 lb ub lb ubsystem variables through which the operational con- such that Tci is some desired value and thatstraints of one affect another. Ts ∈ [Ts , Ts ] is as close as possible to the tar- lb ub In this study we choose to fix Tci which, assum- get TST . If no such solution exists the heliostating Ta is also fixed, determines a simple threshold is not operational. Set Fs = 0, FM = Fsupp .Q˙ H < FR UL Ac (Tc − Ta ) below which operation of i We must therefore derive solutions for Fs in termsthe collector ceases. Moreover, if we assume it is of our control variate TM SF , our operational param-possible to insulate the HTF at this point then we eters Ts , Tc and our system parameters cp etc. Re- oobviate the need to model start-up conditions (which arranging 5 we haverequires non-equilibrium analysis) since the temper-ature at the beginning and end of the day is simply Mc cp (Tco − Tci ) ˙Tci . Fs = , ∆Hsat + ∆Hv + ∆Hs 11
  • b bthe numerator is the rate of heat energy input to Multiplying both sides by c e c y gives an equation ofthe unfired boiler with the denominator the per unit the formenergy output, so that the flow rate is the number ofunits per second required to equate the input with WeW = K (9)the output energy. We note that from 3 we have b b b [ ] where W = c y and K = c e c . Thus we have demon- Tco − Ts Ub Ab strated the equivalence of (8) and (9) whose solutions ln = . Tci − Ts ˙ Mc cp are given by the “Lambert Function” W(·) which do Ub Ab not exist in a simple form (but are rather given as anExponentiating and defining α = e Mc cp we have a infinite sum of simple functions). Therefore we can- ˙linear relation between the unfired boiler input and not complete our expression for Fs (TM SF , Ts , Tci ) ex-exit temperatures plicitly and must make recourse to numeric methods to find solutions. Tco = (1 − α)Ts + αTci . (6) Boiler Performance: Numerical ResultsThe implication being that the difference Tci − Ts isproportionally less than the difference Tco − Ts with Prior to any numerical analysis we must determinethe proportion a non-linear function of the heat ex- the values for system parameters typical for solarchanger characteristics and the heat capacity of the thermal plants. Since we do not have direct access toHTF. Since α = e(x) ∈ [1, ∞) for all x ≥ 0 we de- boiler characteristics for solar thermal installationsduce that (for fixed Tci ) as the proportion α tends to we make use of quoted values for overall heat trans-infinity Ts tends to its upper bound Tci . Therefore fer coefficient Ub = 6780W/m2 for a pressurized wa-we require the receiver input temperature must be ter reactor steam generator given in Table 15.6 Fraasgreater than the target FF temperature so that the (1989) [26]. The operational characteristics thereintarget is reachable. Substituting Tco into our expres- are both commensurate with the temperature pro-sion for Fs gives file of receiver systems such as that given in [2] and with the turbine characteristics for a PWR reactor in Mc cp (1 − α)(Ts − Tci ) ˙ Fs = . (7) Zhou and Turnbull (2002) [27] reproduced in Table ∆Hsat + ∆Hv + ∆Hs 1. Specific heat capacities for the HTF are calculatedSo far we have related the transfer rates in the boiler asto each other but we have yet to equate to the energy cp = 4.6272 × 10−4 T 2 − 0.5805T + 1436.71flux from the receiver system. We note that from (4)and (6) Qc = Mc cp (Tco − Tci ) = Mc cp (1 − α)(Ts − ˙ ˙ ˙ given in [15].Tci ) with M˙ c implicitly a function of Ts . Ideally we ˙ For a given Ts , Qc , Tci and cp we derive numericalwould like to invert this function to obtain an explicit ˙ solutions for Mc from which we derive Fs and Tco . ˙ ˙expression for Mc (Ts , Qc , Tci , cp ), unfortunately it is Our permissible solution set must satisfy operationalnot possible to obtain a simple closed-form solution: constraints on flow rates and temperatures.we note that the aforementioned has the general form Figure 6 (top) illustrates a solution curve for Ts as b ˙ a function of Qc and Fs and the corresponding values Mc (1 − e Mc ) = c ˙ ˙ (8) ˙ for Mc . Inspection confirms our intuition that the Ub Ab ˙ Qc FF temperature increases with increasing flux andwhere b = and c = cp (Ts −Tci ) . Defining y = cp that the temperature decreases as the FF flow-rate1 − M it follows that M c ˙c b ˙c = b(1−y) and substituting c increases. We also make the following observations:into the above we have b(1−y) • The graphs show solutions in the region Mc ∈˙ b −cy b ˙ c reached as y=e c =e e c . [0, 200] with the upper bound for M 12
  • Stage Temperature Pressure Wetness (◦ C) (barA) (%) 1 283 66.77 0.3 Exhaust 157 5.7 15.3 Table 1: Sizewell B PWR High-Pressure Turbine Param- eters, from Zhou and Turnbull (2002) [27]. flux increases one must increase the FF as the HTF flow rate tends to zero when there is too much energy in the HTF for the FF to absorb and maintain the target Tci . • For a given flux and feed-flow rate, decreasing HTF flow increases feed-flow temperature — as ˙ Mc decreases more heat per unit HTF is trans- ferred to the feed-flow thus Ts must increase. Boiler and MSF Interaction In our operational scheme the back-feed wet steam from the turbine is input to the condenser into which cold brine is also fed. At all operation times the brine serves to condense the steam to that it may be pumped back to the unfired/gas boiler to be re-Figure 6: Feed and HTF flow rate solutions as functions heated. The temperature and flow rates of the brineof feed-flow temperature and collection rates. Solutions may also be modulated to provide additional coolingshown for Tci = 288◦ C, Tco = 588◦ C, Ab = 6.9m2 , Mc ≤ of the condensate so as to regulate the temperature ˙200◦ C. (and hence flow-rate) of the FF. Under the opera- tional scheme outlined this allows us to: • control peak electricity production, flux decreases below some minimal value. The ˙ bounds for Ts correspond to those for Mc . • allow the heliostat to operate in low irradiance conditions. • For a fixed feed-flow the HTF mass flow in- creases with decreasing flux — increasing the In the latter case we note that the upper bounds flow rate reduces the residence time of the HTF on the receiver output temperature and HTF mass- in the exchanger, reducing the energy transfer flow rates would otherwise limit the operation time per unit mass and thereby maintaining Tci at for the heliostat under low irradiance conditions. To the target temperature. show this we consider Equation 4 along a Tco -isoline. ˙ Rearranging, it follows that Mc varies linearly with • Conversely HTF flow rates decrease as flux in- ˙ Qc since all solutions are given for fixed Tci . Hence creases so that more energy is absorbed by the Ub Ab feed-flow per unit HTF flow. Eventually as the ˙ ˙ as Qc tends to zero, so does Mc with α = e Mc cp ˙ 13
  • increasing and tending to infinity. Therefore from 5.5(6) Tc Tco − αTci o lim Ts = lim = Tci . Mc Qc →0 α→∞ (1 − α) 4.125 Qc (W[× 10 ])So for Qc decreasing on the line, Ts increases. 7 Similarly for an Mc -isoline we have from (4) that 2.75 Qc lim Tco = lim + Tci = Tci Qc →0 ˙ c cp Qc →0 M 1.375which from (6) implies that 0.0275 Tc − αTci 0.1250 6.2500 12.5 18.750 25 lim Ts = i = Tci Fs (Kg/s) Qc →0 (1 − α) 148.5so that again Ts increases to its maximum along the 148isoline as Qc tends to zero. Thus Ts increases as Qc decreases on the 147.5isolines corresponding to the operational bounds C 147 ˙ lb ˙ ub lb ub(Mc , Mc ] = (0, 150] and (Tco , Tco ] = (0, 650]. ° MSFThe intersection of the regions delineated by these 146.5 Tbounds correspond to a permissible operational re- 146gion through which isolines for Ts will pass. The interval on which the TsT -isoline passes 145.5through this region restricts the set solutions that 145 0 6 12 18 24satisfy our constraints. As can be seen in Figure 7 hour(top) the bounds restrict the permissible region to avery narrow band of Qc , Fs combinations, which re- Figure 7: Permissible region (top) delimited by uppersult in a restricted set of solutions hitting the target ˙ and lower bounds on Tco (red) and Mc (green). Con-TsT . denser rate variation (bottom), no regulation except when In particular for low irradiance conditions no solu- plant operational at low irradiances.tions will exist since all solutions give steam temper-atures above the target 3 . To extend the operation ofthe plant we lower the condenser exit temperature,which for a fixed feed-flow rate reduces the unfiredboiler exit temperature so as to attain the desired which case the condenser temperature is modulatedvalue. This is demonstrated in Figure 7 (bottom) with the boiler exit temperature fixed. Thus con-which shows MSF regulation required on a typical trol of the MSF condenser permits extended runningday n = 100. times and functions as a means to dynamically allo- A similar strategy may be employed should one cate energy flows to electricity and water production.wish to regulate the feed-flow rate to the turbine, in As an addendum we note that as yet the boiler 3 Notethat the set of convex linear sums of the upper and area has not been determined. For the purposes oflower bound of the permissible region cover the region between this study we chose Ab so as to maximise operationthe bounds, and that for each such element we may also saythat limQc →0 Ts = Tci . Thus since Tci > TsT there will exist time by maximising the intersection between the per-a region enclosing the origin for which no solutions hit the missible region and the TST -isoline. A crude approx-target. imation puts this value at Ab = 74m2 . 14
  • Desalination and Agriculture removal technology may be employed but may add substantially to the cost of production — YermiyahuIn many arid and semi-arid regions of the world pre- (2007)[7] estimate that purification and enrichmentcipitation levels are insufficient to meet the needs of costs may double the cost for smaller plants.agricultural production. In such regions farmers mustrely on secondary sources of fresh water. The wide-spread and intensive use of ground-water for human Multi-Stage Flashconsumption and agriculture in many states in India We consider the equilibrium model for MSF systemshave led to a rapid reduction in water-table levels. as presented by Helal (1986)[17] and reproduced inIn northern states such as Gujarat water levels are many studies since ([18],[19],[24],[25] etc.). Our ap-believed have reduced by 26% [5] within the last 15 proach is to derive solutions to the model that equateyears, and in some areas, such as Mehsana are declin- the energy required to attain the target FF conden-ing at rates ranging from 0.91m to 6.02m per year [6]. sate temperature T M SF to the energy transferred to As water levels recede the quality also deterio- the brine. We do not reproduce the equations here asrates, with an estimated 27% [5] of villages in Gu- complete details can be found in the aforementionedjarat having problems associated with excess fluoride publication. We do however outline the approach in-and nitrate levels in ground water — which is the brief.main source of the region’s drinking water [5]. Asidefrom the consequences to human health, high levels MSF Model Overviewof salinity in irrigation water stress plant systems [14]impacting crop yields [13] and soil permeability [14]. Multi-stage flash is a process by which brine is heated Seawater desalination is increasingly seen as a fea- and passed through a series of chambers j = 1, . . . , Nsible solution to global water resource issues and having successively lower pressures so that at eachworld-wide capacity has more than doubled from 12.3 stage the brine is above its saturation temperature,to 35.6 million m3 /day between 1994 and 2004 [7] to causing it to become superheated and flash. Themeet this demand. These plants are often powered flash vapour, virtually free from salt content, con-by fossil fuels and unless alternatives are found will denses on a tube bundle situated at the top of eachhave an increasing impact greenhouse gas emissions chamber through which the cold brine is passed —going forward. MSF and Multi-effect desalination thereby enhancing condensation and pre-heating the(MED) may be configured in co-generation plants cold brine before reaching the boiler. In this way highto make use of waste heat from power systems al- efficiencies can be attained since much of the heat ofthough even in these instances there is some debate vapourisation is reclaimed. Thus the MSF process([21],[22])as to its relative competitiveness with re- uses much less energy than required to simply boilspect to Reverse Osmosis (RO). Although the flow the brine.rates in RO are typically much greater than for MSF In many installations part of the concentratedand MED, RO plants require relatively large amounts brine BN exiting the final chamber is recycled R, soof electricity; being its principal energy requirement. as to make up a percentage of the total input brineThus it is likely the carbon footprint of RO will not W . This can improve efficiency since the heat in thecompare favourably. Additionally it seems that RO brine is re-used.may have limited use in agriculture since the prod- To determine the overall desalination rate DNuct contains relatively high levels of Boron, which Helal assumes equilibrium in each stage, giving rise topasses easily though the semi-porous membrane. Tri- a set of energy, mass and salt conservation equations.als using irrigation water from the world’s largest These equations relate the operational conditions inRO plant in Ashkelon, Israel have indicated that lev- each chamber to empirical properties of brine, waterels of 2mg/litre typical of RO units is “toxic to all and steam and to the heat transfer properties of thebut the most tolerant crops” [11] (in [7]). Boron tube bundle. These relations are typically non-linear 15
  • x 10 6 Given TM SF we determine the required change in enthalpy required to condense and cool the turbine 1.6 back-feed flow F F and solve for the corresponding MSF operational values. In our analysis we assumed 1.4 a similar plant set-up to Helal, partitioning the flash- ing chambers into seven recovery and two rejection D (lb/h) 1.2 chambers. Although no formal optimisation was un- dertaken this combination appeared to give the most N 1 solutions in the operational region for the plant. Fig- 0.8 ure 8 shows a scatter plot of solutions where desali- nation rate is shown against enthalpy for a range of 0.6 operational conditions R, W and CW — the latter being the portion of feed-brine that is rejected with- 1 2 3 4 5 h (J/s) 7 x 10 out entering the recovery section (see Helal for more detail). For any given enthalpy it is clearly optimalFigure 8: Fixed-point solution desailiation rates as a to choose an extremal solution from the set as is in-function of enthalpy, efficient frontier shown in red. dicated by solutions highlighted red. Crop Yield and Irrigation Modeland so the task of solving these simultaneous equa-tions is not trivial. Many factors may affect crop performance within To summarise, the approach is to linearise the over- a given year. Here we are interested in establish-all enthalpy equations about some nominal temper- ing a relationship between yield and irrigation. Weature so that we re-express the original non-linear adopt the meta-approach of Jalota (2007) which con-model as the sum of a linear part and the remain- denses simulation and field results across a vari-ing non-linear part. In Helal’s formulation the linear ety of conditions in the Indian Punjab to generatepart takes the form of a tri-diagonal matrix, which is a generic ‘meta-equation’ whose parameters dependeasily inverted — thus by making a guess for the ini- one of three categorisations of the stage of rainfalltial operational state (temperatures, concentrations, within a given year.mass-flow rates) we may derive a new set of tem- The equation for yield y (kg/ha) is a simpleperatures and hence a new set of concentrations and quadratic function of irrigation QIW (mm/ha)mass-flow rates. These values are used to update ourguess, thereby defining an iterative process. Con- y = a2 Q2 + a1 QIW + a0 + ϵ IW (10)vergence corresponds to a set of operational valuesthat are consistent with the full non-linear equilib- which represents the salient features of the relation-rium equations. ship — namely that a local maximum exists at some intermediate irrigation level below which plants areSolving For Feed-Flow Rates water-deficient and above which water-logging oc- curs. These responses will of course depend on manyAs per our operational scheme we would like to con- factors including soil type, substrate type, topologi-trol Fs to be as as close as possible to the target rate cal relief and other factors that may impact drainage,FF. In instances where Fs would otherwise be greater nutrient and contaminant levels and husbandry fac-than the target from our solution (7) we determine a tors such as sowing density, irrigation timing etc. Thetarget TM SF ≥ TM SFlb = 100◦ C which is above the values for the parameters for (winter) Wheat are re-quoted top-brine temperature TB0 = 93◦ C. produced in Table 2. 16
  • Rainfall State a0 a1 a2 R2 poor -1994 27.24 -0.0274 0.99 below average -2302 37.52 -0.0454 0.99 average -275 36.23 -0.0539 0.97Table 2: Irrigated water yield meta-equation, reproducedfrom Jalota (2007)[8].Biomass Yields and FermentationRatesTo determine biogas production rates we first trans-late grain yields to dry biomass values and map to Figure 9: Electricity Production Rates as a function ofthe final value via appropriate fermentation rates. time of day and year. Assumed turbine efficiency 23.05We assume biomass to grain dry weight ratio of %.1+1.8 1.8 = 1.56 [11] (although we note harvest indicesfor winter wheat in the UK are lower [12]) and usepublished fermentation rates of for wheat straw of and281(litres/kg) given by [9]. These values are unas-sisted rates that do not require extraneous heat in- ∆Hin = (1 − ξi )hf g + SW Tsat +puts — it may be possible in future studies to con- [ξi SW + (1 − ξi )Sv ](TS − Tsat )sider the effect of enhanced gas production in ther-mophilic fermentation in which some of the residual with TSo the exit temperature of the steam as givenheat from the MSF reject flow is used. in Table 1 and all enthalpies, specific heats and satu- ration temperatures appropriate to the temperaturesElectricity Production and pressures given therein. Figure 9 shows electricity production rates as aWe determine electrical energy production rate as function, overnight flow rate translates to 75kW pro-the difference ∆He between the enthalpies of the in- duction. We note that for the given turbine specifi-put and output steam flows to the turbine multiplied cation we estimate efficiencies of approximately 23%,by the feed-flow rate and a thermal efficiency factor which is significantly lower than commonly quotedηT ∈ [0, 1] which is derived from a polynomial fit of values 38 − 40% for tower systems. In later analy-efficiency curves as given in [29] where ses we will consider higher temperature applications more appropriate to tower/reviever systems. ηT = a4 T 4 + a3 T 3 − a2 T 2 + a1 T + a0with a4 = −1.0351e − 013, a3 = 5.6548e − 010, a2 = System Performance under Pos-1.151e − 006, a1 = 0.0010898 and a0 = 0.0021289.Hence itive Feedback Pe = ηT F F ∆He . To demonstrate the effect of the positive feedback mechanism we consider the following operationalHere we calculate ∆He = ∆Hin − ∆Hout with scheme: ∆Hout = (1 − ξo )hf g + SW Tsat + • Assume no extraneous gas support. No gas re- [ξo SW + (1 − ξo )Sv ](TSo − Tsat ) serves in year 0. 17
  • • Operate only when minimum flow rate Fsupp is 365 reachable. • Use gas produced in year i − 1 to support plant 270 6% in year i. 35% day of year 31% 27% Our aim is to derive biogas production and bio- 180 21%gas consumption curves as a function of irrigation 14% 17% 10%level on a fixed area of agricultural land. Via the 6%above iterative scheme the gas produced one year is 90consumed in the next, thereby increasing operationtime and gas consumption. Year-on-year we move 0along the curves until such time that gas consump- 0 6 12 18 24tion equates production — this corresponds to an op- hourerational equilibrium point. The iterative increasein water and compost production, crop yield, and Figure 10: Flux isolines delimit operational time-plant operation time which occurs without extrane- domains, percentage figures showing operational time within each domain.ous input to the system demonstrate the existenceand magnitude of the positive feedback mechanismand the tangible benefits conferred. We note that this equilibrium point will always ex- where hG (J/f t3 /s) is the energy content of biogasist, or will be beyond 100% operation time, so long per cubic foot. Biogas often contains between 50 −as there exists some time within a year for which 70% methane with the remainder mostly comprisingthe plant does not require gas support. Here the ini- carbon dioxide with traces of hydrogen sulphide andtial gas consumption is zero but production is posi- water. For this reason the calorific value of biogastive. Since the curves are both increasing functions is typically much less than natural gas — here weof operation time, the curves either intersect at some assume hG = 5.66 × 105 [11]. Ideally we would modeltime greater than the initial or they do not, imply- the boiler efficiency as a function of temperature anding the feedback mechanism fully supports the plant make-up flow rate, time has not permitted us to makethroughout the year. such an analysis and so we take ηB = 0.7 as given in To derive the gas consumption rates we must trans- Table 1 [10].late the supporting energy requirement to an input If insufficient support is available then operationgas flow rate: For any given make-up flow rate FM will be constrained to some portion of the year,the energy required to vapourise and superheat to TS thus we are interested in deriving consumption andthe condensate flow having input temperature TM SF production values over all operational regions ofand saturation temperature Tsat with input (to the the time-domain. While we may choose the time-turbine) wet-fraction ξi ∈ [0, 1]is domains arbitrarily it seems intuitive to define an op- erational domain that maximises the available energy hsupp = FM (∆Hw + ∆Hv + ∆Hs ). to the plant in that time. We consider a concentric set of domains on the region [0, 24]h × [0, 365]d thatwhere ∆Hsat = SW (Tsat −TM SF ), ∆Hv = (1−ξi )hf g ˙ map to quantiles of heliostat flux field QH (h, d).and ∆Hs = ((1 − ξi )Sv + ξi SW )(TS − Tsat ). Assum-ing a fixed secondary boiler efficiency ηB then the Figure 10 shows the flux isolines and the corre-requisite gas flow rate FG (f t3 /s) is sponding proportion of the time-domain contained within it. Figure 11 top and bottom respectively hsupp show the water production and gas consumption rates FG = throughout the year for a support level Fsupp = ηB hG 18
  • 7 x 10 14 12 10 8 biogas ft /yr 3 6 4 2 0 −2 0 50 100 150 200 250 300 350 irrigation (mm/ha) Figure 12: Annual Gas Consumption and Production vs. Irrigation (mm/ha) given total area of 1239 ha. 90% desalinated water used for irrigation. 60% total irrigation water comes from ground-water sources. water sources. Thus for every six and an half units taken from ground-water ten units are applied to the soil. By replacing more than is removed our irrigation scheme has the potential to recharge this dwindling resource and reduce high-salinity levels often foundFigure 11: Desalination (top) and Gas Consumption in soils of semi-arid regions.4 We have not accounted(bottom) Rates as a function of time of day and year. for any potential increase in yield from this benefit,Here Fsupp = 1kg/s, these flow rates can be achieved nor have we considered yield increases that may arisewithout support at peak irradiance levels, maximum reg- from soil-improvement by composting.ulation occurs when TM SF is minimal. In any one year biogas production depends on the overall water production available for irrigation, which itself increases with annual operational time, feed flow rates and with decreasing condenser output1kg/s. Integrating water rates in the region inte- temperature TM SF . Biogas consumption is a non-rior to an isoline gives the annual water production decreasing function of annual operation time, it in-in that domain from which we determine an annual creases with increasing make-up feed flow rates, andirrigation level (for some fixed irrigation area A(ha)), is zero whenever feed flow rates are equal to or exceedthe corresponding crop yield, dry biomass weight and the support level FF.ultimately gas production. We also obtain the an- Using these curves we may demonstrate how feed-nual gas consumption, thus providing a consump- ing back output gas increases irrigation, crop yieldstion and production pair for the time-domain. Fig- and biogas production in the next year: as per ourure 12 plots these values for support Fsupp = 1kg/s 4 This will depend on the leaching fraction (the proportionagainst irrigation level (mm/ha). Here we have as- of water applied to the surface reaching below the root zone)sumed 10% desalinated water is for municipal use which itself will depend on soil type, climatic conditions, evap-with 65% total irrigation water coming from ground- otranspiration requirements of the crop etc. 19
  • Year 1 Year 2 Eqm first year alone. I mm/ha 260 324 327 In year two, the use of the gas produced in year GP 108 f t3 1.2180 1.2824 1.2861 one gives rise to a 25.9% increase in water production GC108 f t3 0 1.2180 1.2824 with the plant in operation 98% of the year. Since DW 105 m3 1.4325 1.7836 1.8034 the irrigated area is undersized with respect to water G 106 kg 6.8188 7.1795 7.1999 production the response to this additional water is C 106 kg 8.3462 8.7878 8.8127 muted with a 5.6% increase. Gas production also e 106 kWh 3.1213 3.5278 3.5508 increases by the same amount and so after year two Top %yr 35 98 100 the system is virtually at equilibrium with the plant in operation throughout the year. The low support flows we have considered requireTable 3: Gas Consumption (GC), Gas Production (GP), T M SF to be minimal for relatively moderate irradi-Irrigation (I) ,Drinking Water (DW) , Grain (G), Com- ance levels, thus increasing desalination rates duringpost (C),Electricity (e), % operation time Top . Fsupp =1kg/s. the day. In conjunction with the gains from the feed- back loop we are able to attain high desalination rates and irrigation levels over a very large area of land.operational scheme we assume no support in thefirst year of operation. Without support the plantthe annual operation time is 35% which produces Referencesenough water to provide 260 mm/ha of irrigation wa- [1] R. C. Gee, Simple Heat Exchange Factors forter. Mapping this irrigation level to the production Steam Producing Solar Systems, Jn Solar Energycurve we see that approximately 122 million ft3 of Engineering, Vol. 106, (1984).biogas can be generated by the end of the first year.If all this gas is consumed in the next year (i.e. we [2] R. K. McMordie, Convection Heat Loss from amap this to the consumption curve) our operation Cavity Receiver, Jn Solar Energy Engineering,time and water production increase to give an irriga- Vol. 106, (1984).tion level of about 324 mm/ha and corresponding gas [3] Purohit, Energy Policy, Vol. 38, Issue 6, pp 3015-production of 128 million ft3 . In repeating this pro- 3029. (2010)cess we approach the equilibrium — as is illustratedby the arrows in Figure 12 [4] D. Gstoehl et al (2010) ”Towards Industrial Solar Table 3 outlines the system outputs without sup- Production of Zinc and Hydrogen — Modellingport in the first year and the progression of the out- and Design of a 100kW Solar Pilot Reactor forputs as the system reaches equilibrium. The desalina- ZnO Dissociation”, 18th World Hydrogen Energytion output is about 1.8 million cubic meters of which Conference 2010 – Proceedings WHEC 2010.we assume 90% is used for irrigation, which itselfcomprises 40% of the total irrigation supply. What is [5] Gujarat Agriculture: An Overview, Departmentnot apparent from these values is that water is used of Agriculture and Co-operation, Govt. of Gu-in the first year and so the yield we see in year one is jarat.the enhanced yield. If this water had not been avail- [6] M. D. Kumar et al, Economic value of groundwa-able our irrigation level would be 156 mm/ha giving ter: case studies from four villages in Banaskan-a grain and compost production of 5.1382 × 106 kg tha, North Gujarat. Resources, Energy and De-and 6.2891 × 106 kg respectively5 . Used as a baseline velopment (2005) vol 2 issue 1, 1-18we therefore see an increase of 40.1% in grain andbiogas production due to additional irrigation in the [7] U. Yermiyahu et al, Rethinking Desalinated Wa- ter Quality and Agriculture. Science (2007) Vol 5 Assuming a compost to crop residue ratio of 0.68. 318. p 920-921. 20
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